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arxiv: 2605.21525 · v1 · pith:YWOXJAIEnew · submitted 2026-05-19 · 🧮 math.GM

Monoidal Alphabets for Generalized Harmonic Sums

Jayanta Phadikar This is my paper

Pith reviewed 2026-05-22 01:48 UTC · model grok-4.3

classification 🧮 math.GM
keywords monoidal alphabetsgeneralized harmonic sumsEuler sumsstuffle algebramultiple harmonic numbersaffine harmonic numberspolynomial-base sums
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The pith

Monoidal alphabets unify generalized Euler sums by closing under pointwise multiplication to induce a stuffle algebra on nested sums.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a framework for organizing Euler-type sums around the idea of a monoidal alphabet, a collection of summand functions closed under pointwise multiplication. This closure property induces the standard stuffle product on the associated nested sums, placing classical multiple harmonic numbers, colored sums, and various generalized versions inside the same algebraic setting. The author concentrates on three concrete families: sums built from powers of n, from linear factors an plus b, and from polynomial factors in n. Closure and lifting theorems then show that nested sums formed from these families, even when multiplied by ordinary harmonic-number factors, can be rewritten as finite combinations of the corresponding harmonic-number objects.

Core claim

We develop a general finite-alphabet framework for Euler-type sums based on the notion of a monoidal alphabet. An alphabet of summand letters is called monoidal when it is closed under pointwise multiplication, thereby inducing the usual stuffle, or quasi-shuffle, algebra on the associated nested sums. This viewpoint places classical multiple harmonic numbers, colored harmonic sums, and several generalized Euler sums under a common structural mechanism. We focus on three fundamental families of monoidal alphabets: the ordinary power alphabet generated by n, the affine alphabet generated by linear factors an+b, and the polynomial-base alphabet generated by polynomial factors P(n). The paper's

What carries the argument

A monoidal alphabet: any set of functions from positive integers to numbers that is closed under pointwise multiplication, thereby equipping the nested sums built from those functions with a stuffle algebra.

If this is right

  • Nested sums from the power, affine, and polynomial-base alphabets reduce to finite harmonic-number objects.
  • Many previously known Euler-sum identities are recovered in a uniform algebraic manner.
  • New identities for generalized sums are generated systematically without ad-hoc case analysis.
  • The reduction step to simpler closed-form expressions is separated from the structural closure step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same monoidal-alphabet construction could be tested on alphabets built from other simple arithmetic progressions or from exponential factors.
  • Automatic identity generators could be built by enumerating monoidal alphabets and applying the lifting rules.
  • The framework may clarify which classes of sums remain irreducible even after the closure step is performed.

Load-bearing premise

The chosen sets of summand functions must be closed under pointwise multiplication at each positive integer.

What would settle it

Exhibit one concrete nested sum whose summands come from a monoidal alphabet (power, affine, or polynomial-base) together with possible harmonic-number multipliers, yet cannot be expressed as a finite linear combination of the corresponding finite harmonic numbers.

read the original abstract

We develop a general finite-alphabet framework for Euler-type sums based on the notion of a monoidal alphabet. An alphabet of summand letters is called monoidal when it is closed under pointwise multiplication, thereby inducing the usual stuffle, or quasi-shuffle, algebra on the associated nested sums. This viewpoint places classical multiple harmonic numbers, colored harmonic sums, and several generalized Euler sums under a common structural mechanism. We focus on three fundamental families of monoidal alphabets: the ordinary power alphabet generated by $n$, the affine alphabet generated by linear factors $an+b$, and the polynomial-base alphabet generated by polynomial factors $P(n)$. The resulting classes of multiple harmonic numbers, multiple affine harmonic numbers, and multiple polynomial-base harmonic numbers provide systematic containers for a wide range of finite and infinite Euler-type sums. We prove closure and lifting results showing that nested sums whose summands are built from these alphabets, possibly multiplied by harmonic-number factors, reduce to the corresponding finite harmonic-number objects. As consequences, the framework recovers many known Euler-sum identities and produces many new identities in a uniform way. While reduction to simpler functions remains a separate and often difficult problem, the monoidal-alphabet perspective provides a unified algebraic language for organizing, transforming, and extending harmonic-sum identities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a monoidal-alphabet framework for Euler-type sums, where an alphabet is monoidal if closed under pointwise multiplication, thereby inducing the stuffle algebra on associated nested sums. It treats three families—power alphabets generated by powers of n, affine alphabets generated by linear factors an+b, and polynomial-base alphabets generated by polynomials P(n)—and proves closure and lifting results asserting that nested sums built from these alphabets (possibly multiplied by harmonic-number factors) reduce to the corresponding finite harmonic-number objects. The framework is presented as recovering known identities and generating new ones uniformly.

Significance. If the monoidality assertions and the closure/lifting proofs hold for all three families, the work supplies a uniform algebraic language for organizing generalized harmonic sums. The explicit separation of the reduction step from the subsequent simplification to simpler functions is a constructive feature. The approach could systematize identities across colored, multiple, and generalized Euler sums, provided the foundational definitions are free of gaps.

major comments (1)
  1. [Definition of the affine alphabet and monoidality] The definition of the affine alphabet (linear factors an+b) and the claim that it is monoidal appear in the section introducing the three families. Pointwise multiplication of two generators yields a quadratic term ac n^2 + …, which lies outside the linear class. Unless the alphabet is redefined as the monoid generated by all such products (i.e., a suitable class of polynomials), the stuffle product of two affine summands does not remain inside the multiple affine harmonic numbers. This directly affects the applicability of the closure and lifting theorems to the affine family; the manuscript must either supply the explicit closure proof or adjust the statement of the central results.
minor comments (1)
  1. Notation for the finite versus infinite objects could be made more uniform; a short table contrasting the three alphabet families with their induced sum classes would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: The definition of the affine alphabet (linear factors an+b) and the claim that it is monoidal appear in the section introducing the three families. Pointwise multiplication of two generators yields a quadratic term ac n^2 + …, which lies outside the linear class. Unless the alphabet is redefined as the monoid generated by all such products (i.e., a suitable class of polynomials), the stuffle product of two affine summands does not remain inside the multiple affine harmonic numbers. This directly affects the applicability of the closure and lifting theorems to the affine family; the manuscript must either supply the explicit closure proof or adjust the statement of the central results.

    Authors: We agree that the set of individual linear factors an+b is not closed under pointwise multiplication, as the product of two such factors is generally quadratic. The manuscript intends the affine alphabet to be the monoid generated by these linear factors under pointwise multiplication; this monoid consists of all finite products of linear terms and therefore comprises a suitable class of polynomials. We will revise the introductory section to state this definition explicitly, include a short direct proof that the monoid is closed by construction, and confirm that the stuffle algebra and the closure/lifting theorems therefore apply unchanged to the resulting multiple affine harmonic numbers. This clarification also aligns the affine case with the polynomial-base family. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard algebraic definitions and proofs.

full rationale

The paper introduces monoidal alphabets by explicit definition (closed under pointwise multiplication to induce the stuffle algebra) and then states theorems for closure and lifting of nested sums built from power, affine, and polynomial-base families. These steps are presented as consequences of the definition plus standard properties of quasi-shuffle products; no equation or result is shown to reduce to a prior fitted parameter, self-referential equation, or load-bearing self-citation whose content is itself unverified. The affine family is asserted to be monoidal, and while this assertion may raise separate questions of correctness if the generators are strictly linear, the logical structure does not exhibit any of the enumerated circular patterns. The overall argument remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central construction rests on the definition of monoidal alphabets and the induced stuffle algebra; no free parameters or new physical entities are introduced.

axioms (1)
  • domain assumption An alphabet is monoidal when closed under pointwise multiplication, thereby inducing the stuffle algebra on nested sums.
    This is the foundational definition stated in the abstract that enables all subsequent closure and lifting results.

pith-pipeline@v0.9.0 · 5748 in / 1264 out tokens · 46217 ms · 2026-05-22T01:48:23.451674+00:00 · methodology

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Reference graph

Works this paper leans on

80 extracted references · 80 canonical work pages

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    =H (i) k (2−k H(2) k (−1 3 −i)−2H (2) k (−1 6 − i 2))−2H (i,2) k (1 2 ,− 1 3 −i) + 2H (i,2) k (1,− 1 6 − i 2) Identity 4 kX n=1 χ3,2(n)H (1,2) n (s1, s2) = 1 3 (−1) 1 3 (1 + (−1) 1 3 ) (H(1,2) k (−((−1) 1 3 )s 1, s2)− H (1,2) k ((−1) 2 3 s1, s2) +H (0,1,2) k (−((−1) 1 3 ), s1, s2)− H (0,1,2) k ((−1) 2 3 , s1, s2)) Identity 5 kX n=1 Mod (n,3)H(1,2) n Hn = ...

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    e 2 ik π 3 + 2k (5 i + √

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    e 4 ik π 3 −2 (3 2k √ 3 + i (−1−i √ 3)k + 3 2k √ 3k))H (1,2) k + 21+k √ 3 (6 +H (1,2) k (−1 2 − i √ 3 2 ,1) +H (1,2) k (1 2 i (i + √ 3),1))) + √ 3 (H(2) k (3 + 12k+ 2H (2) k (−1 2 − i √ 3 2 ) + 2H (2) k (1 2 i (i + √ 3)))−2 (H (4) k (−1 2 − i √ 3 2 ) +H (4) k (1 2 i (i + √ 3)) + 3H (1,2) k + 3kH (1,2) k +H (1) k (−1 2 − i √ 3 2 )H (1,2) k +H (1) k (1 2 i ...

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    + 3H(4,1) k (1 3 ,1) + 3H (2,0,3) k (1,1, 1

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    +H (2,0,0,2,1) k (1,1,1, 1 3 ,1) +H (2,0,2,0,1) k (1,1, 1 3 ,1,1) +H (2,2,0,0,1) k (1, 1 3 ,1,1,1) 3 Identity 8 kX n=1 Fn,3 (H(2 i) n )2 n 2 = 1 4 √ 13 (−4A (4 i) k (1 2 (−3 + √ 13)) + 2A (4 i) k (−3 + √

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    (1 + 1 2 (−3 + √ 13)) − (3− √ 13)1+k H(4 i) k 2− √ 13 + (3 + √ 13)1+k H(4 i) k 2 + √ 13 + 2 (−(1

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    (3− √ 13)A (4 i) k ( 1 2 (−3 + √ 13))−( 1 2 (3− √ 13))1+k H(4 i) k ) 1 + 1 2 (−3 + √ 13) − −(3− √ 13)A (4 i) k (−3 + √ 13)−(3− √ 13)1+k H(4 i) k −2 + √ 13 + 2−k (3− √ 13)1+k H(4 i) k (2) −1 + 1 2 (3− √

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    (3− √ 13)A (4 i) k (−3 + √ 13)−( 1 2 (3− √ 13))1+k H(4 i) k (2) 1 + 1 2 (−3 + √

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    (−(3− √ 13)A (4 i) k (1 2 (−3 + √ 13))+ 1 2 (3− √ 13)A (4 i) k (−3+ √ 13)−2 −k (3− √ 13)1+k H(4 i) k + (1 2 (3− √ 13))1+k H(4 i) k (2))−4H (4 i) k (1 2 (3+ √ 13)) − 2 (−(( 2 3+ √ 13)−1−k)H (4 i) k + 1 2 (3 + √ 13)H (4 i) k ( 1 2 (3 + √ 13))) 1 + 1 2 (−3− √

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    (1 + 1 2 (−3− √ 13)) + 1 −1 + 1 2 (3 + √

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    (−(2−k) (3 + √ 13)1+k H(4 i) k + (1 2 (3 + √ 13))1+k H(4 i) k (2) + (3 + √ 13)H (4 i) k (1 2 (3 + √ 13))− 1 2 (3 + √ 13)H (4 i) k (3 + √ 13))+ −(( 2 3+ √ 13)−1−k)H (4 i) k (2) + 1 2 (3 + √ 13)H (4 i) k (3 + √ 13) 1 + 1 2 (−3− √

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    + −((3 + √ 13)1+k)H (4 i) k + (3 + √ 13)H (4 i) k (3 + √ 13) −2− √ 13 + 4H (2 i,2 i) k (1 2 (3− √ 13),1)−2H (2 i,2 i) k (1 2 (3− √ 13),2)−2H (2 i,2 i) k (3− √ 13,1)−4H (2 i,2 i) k (1 2 (3 + √ 13),1) + 2H (2 i,2 i) k (1 2 (3 + √ 13),2) + 2H (2 i,2 i) k (3 + √ 13,1) + 4H (0,2 i,2 i) k (1 2 (3− √ 13),1,1)−2H (0,2 i,2 i) k (1 2 (3 4 − √ 13),1,2)−2H (0,2 i,2 i...

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    + (30821 + 713 i √ 119) (4 i (3 i + √ 119))k)H (1,2) k + (119−515 i √ 119)H (1,2) k (−12−4 i √ 119,1) + (119 + 515 i √ 119)H (1,2) k (−12 + 4 i √ 119,1)) Identity 13 kX n=1 ( n 15)H (1,2) n = −1√ 15 i (H(1,2) k (−((−1) 1 15 ),1) +H (1,2) k ((−1) 2 15 ,1) +H (1,2) k ((−1) 4 15 ,1)− H(1,2) k (−((−1) 7 15 ),1) +H (1,2) k ((−1) 8 15 ,1)− H (1,2) k (−((−1) 11 ...

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    +H (0,1,2) k (−((−1) 1 4 ),1,1)− H (0,1,2) k ((−1) 1 4 ,1,1)− H (0,1,2) k (−((−1) 3 4 ),1,1) +H (0,1,2) k ((−1) 3 4 ,1,1)) 6 Identity 15 kX n=1 Quotient (2n+ 1,5)H (1,2) n = 1 25 (10 (−Hk +kH (2) k )−5H (1,2) k + 10H (−1,1,2) k −5H (0,1,2) k − H(1,2) k (−((−1) 1 5 ),1) + 4 (−1) 1 5 H(1,2) k (−((−1) 1 5 ),1)−2 (−1) 2 5 H(1,2) k (−((−1) 1 5 ),1) −3 (−1) 4 5...

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    + 4 (−1) 3 5 H(1,2) k (−((−1) 3 5 ),1)− H (1,2) k ((−1) 4 5 ,1) + 3 (−1) 1 5 H(1,2) k ((−1) 4 5 ,1) + 2 (−1) 3 5 H(1,2) k ((−1) 4 5 ,1)−4 (−1) 4 5 H(1,2) k ((−1) 4 5 ,1)− H (0,1,2) k (−((−1) 1 5 ),1,1) + 4 (−1) 1 5 H(0,1,2) k (−((−1) 1 5 ),1,1)−2 (−1) 2 5 H(0,1,2) k (−((−1) 1 5 ),1,1) −3 (−1) 4 5 H(0,1,2) k (−((−1) 1 5 ),1,1)− H (0,1,2) k ((−1) 2 5 ,1,1)−...

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    ((−5−5 i)H (1,2) k −(5 + 5 i)H (0,1,2) k + (1 + 2 i)H(1,2) k (−i,1) + (2 + i)H (1,2) k (i,1) + (1 + 2 i)H(0,1,2) k (−i,1,1) + (2 + i)H(0,1,2) k (i,1,1)) 7 Identity 18 kX n=1 ν4(12 2n)H (1,2) n = 1 4 (2 (−Hk +kH (2) k ) + 3H (1,2) k + 2H (−1,1,2) k + 3H (0,1,2) k +H (1,2) k (−1,1) +H (0,1,2) k (−1,1,1)) Identity 19 kX n=1 σ2(2n)H (1,2) n = 1 3 (−H(1,2) k −...

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    +H (3,2) 3+2k(1 4 , 1 5)− H (2,1,2) 3+2k (−1, 1 4 , 1

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    +H (2,1,2) 3+2k (1, 1 4 , 1 5)) Identity 22 kX n=1 (2n+ 5) 2 i (h[1] 2n+5(1; 1))2 =−1−169 3 −2+2 i − 7569 4 5−2+2 i + 1 2 A(2−2 i) 5+2k − 1 2 ζ(2−2 i,6 + 2k) + 2H (−2 i,2) 5+2k + 2H (1−2 i,1) 5+2k + 5 2 H(−2 i,0,2) 5+2k 8 + 6H (−2 i,1,1) 5+2k +H (1−2 i,0,1) 5+2k +H (−2 i,0,0,2) 5+2k + 6H (−2 i,0,1,1) 5+2k + 2H (−2 i,1,0,1) 5+2k + 2H (−2 i,0,0,1,1) 5+2k +H...

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    +H (2,2,1) 5+2k (−1, 1 3 ,−1)− H (2,2,1) 5+2k (1, 1 3 ,−1)) Identity 24 kX n=1 ( 1 2)n H2n+3 H(1,2) 2n+3 (2n+ 3) 2 = √ 2 (− 121 648 √ 2 − H(3,3) 3+2k(−( 1√ 2),1) +H (3,3) 3+2k( 1√ 2 ,1)− H (4,2) 3+2k(−( 1√ 2),

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    +H (4,2) 3+2k( 1√ 2 ,1)− H (2,1,3) 3+2k (−( 1√ 2),1,1) +H (2,1,3) 3+2k ( 1√ 2 ,1,1) − H(2,2,2) 3+2k (−( 1√ 2),1,1) +H (2,2,2) 3+2k ( 1√ 2 ,1,1)−2H (3,1,2) 3+2k (−( 1√ 2),1,1) + 2H (3,1,2) 3+2k ( 1√ 2 ,1,1)− H (3,2,1) 3+2k (−( 1√ 2),1,1) +H (3,2,1) 3+2k ( 1√ 2 ,1,1)−2H (2,1,1,2) 3+2k (−( 1√ 2),1,1,1) + 2H (2,1,1,2) 3+2k ( 1√ 2 ,1,1,1)− H (2,1,2,1) 3+2k (−(...

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    (−2 + 2 i +Hk (−3 i ik A(i+ √ 6) k (2 3)−3H (i+ √ 6) k (−(2 i 3 ))) + (3 + 3 i)H(1+i+ √ 6) 1+k (−(2 i 3 ))−3H (1,i+ √ 6) k (i,−( 2 3)) + 3H (1,i+ √ 6) k (1,−( 2 i 3 )) + (3 + 3 i)H(1,i+ √ 6) 1+k (i,−( 2 3)) + (3 + 3 i)H(i+ √ 6,1) 1+k (−(2 i 3 ),1)) 10 Identity 28 kX n=1 H(1,2) n (3n−1) 2 =−3H (3) −1+3k + 3 (−1) 1 3 H(3) −1+3k −3 (−1) 2 3 H(3) −1+3k + 3H (...

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    =G k({{−2 i,2},{3}};{−( i 6), i 8 };{{{7,3},{1,0}},{{1,0}}}) + Gk({{−2 i},{2},{3}};{−1, i 6 , i 8 };{{{7,3}},{{1,0}},{{1,0}}}) Identity 33 kX n=1 ( 1 2)n H(2,3) n (i, 7 3) (6n+ 4) i (8n−2) 2 i (10n−8) 3 i =G k({{i,2 i,3 i,2},{3}};{ i 2 , 7 3 };{{{6,4},{8,−2},{10,−8},{1,0}},{{1,0}}}) + Gk({{i,2 i,3 i},{2},{3}};{ 1 2 ,i, 7 3 };{{{6,4},{8,−2},{10,−8}},{{1,0}...

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    =P k({{−1 2 −i,2},{3}};{−( i 6), i 8 };{{{2,3,1},{0,1}},{{0,1}}}) + Pk({{−1 2 −i},{2},{3}};{−1, i 6 , i 8 };{{{2,3,1}},{{0,1}},{{0,1}}}) Identity 37 kX n=1 ( 1 2)n H(2,3) n (i, 7 3) (1 +n+n 2)i (3−2n+ 5n 2)2 i (7 +n 3)3 i =P k({{i,2 i,3 i,2},{3}};{ i 2 , 7 3 };{{{1,1,1},{3,−2,5},{7,0,0,1},{0,1}},{{0,1}}}) + Pk({{i,2 i,3 i},{2},{3}};{ 1 2 ,i, 7 3 };{{{1,1,...

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    + 3H (4,1) 2k (−( 1√ 2),1) + 3H (4,1) 2k ( 1√ 2 ,1) + 3 H(4,0,1) 2k (−( 1√ 2),1,1) + 3H (4,0,1) 2k ( 1√ 2 ,1,1) +H (4,0,0,1) 2k (−( 1√ 2),1,1,1) +H (4,0,0,1) 2k ( 1√ 2 ,1,1,1)) 16 Identity 44 kX n=1 ( 1 2)n H(i,2 i) 2n n4 = 8 (H(4+i,2 i) 2k (−( 1√ 2),1) +H (4+i,2 i) 2k ( 1√ 2 ,1) +H (4,i,2 i) 2k (−( 1√ 2),1,1) +H (4,i,2 i) 2k ( 1√ 2 ,1,1)) Identity 45 kX ...

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    (−1) 1 3 )− H (2) 3k(1 3 (−1) 2 3 ) + 3H (3) 3k(1

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    (−1) 1 3 ) 1−(−1) 2 3 + 3H (3) 3k(1 3 (−1) 2 3 ) + 3kH (3) 3k(1 3 (−1) 2 3 ) + −((−((−1) 1 3 ))1+3k )H (3) 3k(−( 1

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    + (−1) 2 3 H(3) 3k( 1 3 (−1) 2 3 ) 1−(−1) 2 3 + (−1) 2 3 H(3) 3k(−( 1

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  42. [42]

    (−1) 1 3 )−(−1) 2 3 (1+3k ) H(3) 3k( 1 3 (−1) 2 3 ) 1−(−1) 2 3 + −((−1) 1 3 )H (3) 3k( 1 3)−(−((−1) 1 3 ))1+3k H(3) 3k( 1 3 (−1) 2 3 ) 1 + (−1) 1 3 −(−H (1) 3k(1

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  43. [43]

    + 3k H(2) 3k(1 3)) log (3)− (−((−((−1) 1 3 ))1+3k )H (2) 3k( 1 3)−(−1) 1 3 H(2) 3k(−( 1

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  44. [44]

    (−1) 1 3 )) log (3) 1 + (−1) 1 3 − H(2) 3k(1 3 (−1) 2 3 ) log (3)− (−((−1) 2 3 (1+3k ))H (2) 3k( 1

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  45. [45]

    + (−1) 2 3 H(2) 3k( 1 3 (−1) 2 3 )) log (3) 1−(−1) 2 3 + (−H(1) 3k(1

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  46. [46]

    + 3kH (2) 3k(1 3)) log (1 17 + (−1) 1 3 )+ (−((−((−1) 1 3 ))1+3k )H (2) 3k( 1 3)−(−1) 1 3 H(2) 3k(−( 1

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  47. [47]

    (−1) 1 3 )) log (1 + (−1) 1 3 ) 1 + (−1) 1 3 +H (2) 3k(1 3 (−1) 2 3 ) log (1 + (−1) 1 3 ) + (−((−1) 2 3 (1+3k ))H (2) 3k( 1

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  48. [48]

    + (−1) 2 3 H(2) 3k( 1 3 (−1) 2 3 )) log (1 + (−1) 1 3 ) 1−(−1) 2 3 + (−H(1) 3k(1

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  49. [49]

    + 3kH (2) 3k(1 3)) log (1−(−1) 2 3 ) + (−((−((−1) 1 3 ))1+3k )H (2) 3k( 1 3)−(−1) 1 3 H(2) 3k(−( 1

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  50. [50]

    (−1) 1 3 )) log (1−(−1) 2 3 ) 1 + (−1) 1 3 +H (2) 3k(1 3 (−1) 2 3 ) log (1−(−1) 2 3 ) + (−((−1) 2 3 (1+3k ))H (2) 3k( 1

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  51. [51]

    + (−1) 2 3 H(2) 3k( 1 3 (−1) 2 3 )) log (1−(−1) 2 3 ) 1−(−1) 2 3 +H (2) 3k(1

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  52. [52]

    (log (1 3 (1 + (−1) 1 3 )) + log (1−(−1) 2 3 )) +H (2) 3k(−(1

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  53. [53]

    (−1) 1 3 ) (log (1 3 (1 + (−1) 1 3 )) + log (1−(−1) 2 3 )) + 3H(1,2) 3k (1, 1

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  54. [54]

    + 3H(1,2) 3k (−((−1) 1 3 ), 1 3) + 3H (1,2) 3k ((−1) 2 3 , 1

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  55. [55]

    +H (2,1) 3k (1 3 ,1) +H (2,1) 3k (1 3 ,−((−1) 1 3 )) +H (2,1) 3k (1 3 ,(−1) 2 3 ) +H (2,1) 3k (−(1

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  56. [56]

    (−1) 1 3 ,1) +H (2,1) 3k (−(1

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  57. [57]

    (−1) 1 3 ,−((−1) 1 3 )) +H (2,1) 3k (−(1

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  58. [58]

    (−1) 1 3 ,(−1) 2 3 ) +H (2,1) 3k (1 3 (−1) 2 3 ,1) +H (2,1) 3k (1 3 (−1) 2 3 ,−((−1) 1 3 )) +H (2,1) 3k (1 3 (−1) 2 3 ,(−1) 2 3 ) +H (0,1,2) 3k (1,1, 1 3) +H (0,1,2) 3k (1,−((−1) 1 3 ), 1

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  59. [59]

    +H (0,1,2) 3k (1,(−1) 2 3 , 1

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  60. [60]

    +H (0,1,2) 3k (−((−1) 1 3 ),1, 1

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  61. [61]

    +H (0,1,2) 3k (−((−1) 1 3 ),−((−1) 1 3 ), 1

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  62. [62]

    +H (0,1,2) 3k (−((−1) 1 3 ),(−1) 2 3 , 1 3) +H (0,1,2) 3k ((−1) 2 3 ,1, 1

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  63. [63]

    +H (0,1,2) 3k ((−1) 2 3 ,−((−1) 1 3 ), 1

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  64. [64]

    +H (0,1,2) 3k ((−1) 2 3 ,(−1) 2 3 , 1

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  65. [65]

    +H (0,2,1) 3k (1, 1 3 ,1) +H (0,2,1) 3k (1, 1 3 ,−((−1) 1 3 )) +H (0,2,1) 3k (1, 1 3 ,(−1) 2 3 ) +H (0,2,1) 3k (−((−1) 1 3 ), 1 3 ,1) +H (0,2,1) 3k (−((−1) 1 3 ), 1 3 ,−((−1) 1 3 )) +H (0,2,1) 3k (−((−1) 1 3 ), 1 3 ,(−1) 2 3 ) +H (0,2,1) 3k ((−1) 2 3 , 1 3 ,1) +H (0,2,1) 3k ((−1) 2 3 , 1 3 ,−((−1) 1 3 )) +H (0,2,1) 3k ((−1) 2 3 , 1 3 ,(−1) 2 3 )) 18 Ident...

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  66. [66]

    (1 + i q 2 1+ √ 5) + 3 (−((−i)1+2k ) 2 1 2 (1+2k ) (1 + √ 5) 1 2 (−1−2k ) H2k −i q 2 1+ √ 5 H(1) 2k(−i q 2 1+ √ 5)) 1 + i q 2 1+ √ 5 + i1+2k 2 1 2 (1+2k ) (1 + √ 5) 1 2 (−1−2k ) A2k + i q 2 1+ √ 5 H(1) 2k(−i q 2 1+ √ 5) 1−i q 2 1+ √ 5 + 2H (1) 2k(i s 2 1 + √

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  67. [67]

    + 2H (1) 2k(i q 2 1+ √ 5) (1 + √

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  68. [68]

    (1 + i q 2 1+ √ 5) + (−i)1+2k 2 1 2 (1+2k ) (1 + √ 5) 1 2 (−1−2k ) A2k −i q 2 1+ √ 5 H(1) 2k(i q 2 1+ √ 5) 1 + i q 2 1+ √ 5 21 + 3 (−(i1+2k ) 2 1 2 (1+2k ) (1 + √ 5) 1 2 (−1−2k ) H2k + i q 2 1+ √ 5 H(1) 2k(i q 2 1+ √ 5)) 1−i q 2 1+ √ 5 −2H (1) 2k( r 1 2 (1 + √ 5)) + (1 + √ 5)H (1) 2k( q 1 2 (1 + √ 5)) 2 (1− q 1 2 (1 + √ 5)) (1 + q 1 2 (1 + √ 5)) − (−1)1+2...

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  69. [69]

    + 208410602400000ζ(6,4,1,1) + 1289998710000π 6 (ζ(3))2 −8172964800000 (ζ(3))4 −4236320088000π 4 ζ(3)ζ(5) −151947821025000π 2 (ζ(5))2 −387960422850000π 2 ζ(3)ζ(7)−2161493784450000ζ(5)ζ(7) + 1348164597780000ζ(3)ζ(9)) Identity 55 ∞X n=1 An A(2) n n4 = 1 960 (−960 (HPL4,−2,1(1) + HPL4,−1,2(1)) +π 6 log (4)−8π 4 ζ(3)−2200π 2 ζ(5) + 22935ζ(7)) 23 Identity 56 ∞X...

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  70. [70]

    +ζ( 13 2 + 2 i 3 ) Identity 59 ∞X n=1 ( 1 2)n h[1] n (1; 1 2)h [1] n (1; 1) n2−i = 4 Li 2−i,2(1 2 , 1

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  71. [71]

    + 2 Li3−i,1(1 4 ,1) + 2 Li 3−i,1(1 2 , 1

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  72. [72]

    + 5 Li2−i,0,2(1 2 ,1, 1

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  73. [73]

    + 6 Li2−i,1,1(1 2 , 1 2 ,1) + 6 Li 2−i,1,1(1 2 ,1, 1 2) + Li3−i,0,1(1 4 ,1,1) + Li 3−i,0,1(1 2 ,1, 1

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  74. [74]

    + 2 Li2−i,0,0,2(1 2 ,1,1, 1

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  75. [75]

    + 6 Li2−i,0,1,1(1 2 ,1, 1 2 ,1) + 6 Li 2−i,0,1,1(1 2 ,1,1, 1 2) + 2 Li2−i,1,0,1(1 2 , 1 2 ,1,1) + 2 Li 2−i,1,0,1(1 2 ,1,1, 1

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  76. [76]

    + 2 Li2−i,0,0,1,1(1 2 ,1,1, 1 2 ,1) + 2 Li 2−i,0,0,1,1(1 2 ,1,1,1, 1

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  77. [77]

    + Li2−i,0,1,0,1(1 2 ,1, 1 2 ,1,1) + Li2−i,0,1,0,1(1 2 ,1,1,1, 1

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  78. [78]

    + Li4−i(1 4) 24 Identity 60 ∞X n=1 (H(1,2) n )2 n3−6 i =ζ(5−6 i,4) +ζ(3−6 i,2,4) + 2 (ζ(4−6 i,1,4) +ζ(4−6 i,3,2) +ζ(5−6 i,2,2) +ζ(3−6 i,1,1,4) +ζ(3−6 i,1,3,2) +ζ(3−6 i,2,2,2) + 2ζ(4−6 i,1,2,2) +ζ(4−6 i,2,1,2) + 2ζ(3−6 i,1,1,2,2) +ζ(3−6 i,1,2,1,2)) Identity 61 ∞X n=1 ( 1 2)n An A(2) n n4 = HPL4,3(1 2)−HPL 5,−2(−(1 2))−HPL 6,−1(−(1 2)) + HPL−4,1,−2(−(1 2)) ...

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  79. [79]

    + Li7(−(1 2)) + Li7(1 2)) Identity 64 ∞X n=1 A(2) 4n n4 =− 3107π 6 45360 + 64 (HPL−4,2(i) + HPL4,−2(i) + HPL4,−2(1)) + 48 (ζ(3))2 25 Identity 65 ∞X n1 =1 n1X n2 =1 Hn1 Hn2 n2 1 n3 2 =−( 23 180)π 4 ζ(3) + 3 4 π2 ζ(5) + 10ζ(7) Identity 66 ∞X n1 =1 n1X n2 =1 n2X n3 =1 Hn1 Hn2 Hn3 n2 1 n3 2 n4 3 = 169364029π 12 891596160000 + 5567π 6 (ζ(3))2 90720 − π4 (843ζ(...

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  80. [80]

    + Li2,3,2(1 2 , 1 3 ,1) + 2 Li 2,4,1(1 2 , 1 3 ,1) + Li 3,3,1(1 2 , 1 3 ,1) + Li 2,1,3,1(1 2 ,1, 1 3 ,1) + 2 Li2,3,1,1(1 2 , 1 3 ,1,1) + Li 7(1

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